Quantitative Aptitude

Algebra

3,715 Questions
All Polynomial Roots and Coefficients 2451 Zero-Product Root Finding 667 Linear System Manipulation 162 Polynomial Theorems and Reduction 85 Algebraic Identity Simplification 82 Surds and Indices 70 Modulus and Absolute Value 54 Algebraic Expression Substitution 46 Algebraic Fraction Simplification 33 Logarithmic Equations 25 Repeated Series Sum 12 Linear Equation Word Problems 9

Algebra Questions

Multiple choice

State whether True or False If we can factorize $ax^2 + bx + c, a\neq 0$, into a product of two linear factors, then the roots of the quadratic equation $ax^2 + bx + c = 0$ can be found by equating each factor to zero ?

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If a quadratic expression can be factored into (px + q)(rx + s) = 0, then by the zero product property, either px + q = 0 or rx + s = 0. This is the fundamental basis for solving quadratic equations by factoring.

Multiple choice

Determine the nature of roots of the equation $x^2 + 2x\sqrt{3}+3=0$.

  1. Real and distinct

  2. Non-real and distinct

  3. Real and equal

  4. Non-real and equal
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The discriminant of the quadratic equation x^2 + 2x*sqrt(3) + 3 = 0 is D = b^2 - 4ac. Here, a=1, b=2*sqrt(3), and c=3. D = (2*sqrt(3))^2 - 4(1)(3) = 12 - 12 = 0. Since the discriminant is zero, the roots are real and equal.

Multiple choice

The roots of the equation $x^2 - 2x = 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:
  1. $ y = x^2, y = 2x$
  2. $y = x^2 - 2x, y =0$
  3. $y = x, y = x - 2$
  4. $y = x^2 - 2x + 1, y =1$
  5. $y = x^2 - 1, y =2x - 1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The roots of x^2 - 2x = 0 are x = 0 and x = 2. Option C represents the system y = x and y = x - 2, which intersect at x = x - 2, implying 0 = -2, which is impossible. The other options represent systems that intersect at the correct roots.